Multi-peak bound states for Schrödinger equations with compactly supported or unbounded potentials
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 5, pp. 1205-1226.

In this paper, we will study the existence and qualitative property of standing waves $\psi \left(x,t\right)={e}^{-\frac{iEt}{ϵ}}u\left(x\right)$ for the nonlinear Schrödinger equation $iϵ\frac{\partial \psi }{\partial t}+\frac{{ϵ}^{2}}{2m}{\Delta }_{x}\psi -\left(V\left(x\right)+E\right)\psi +K\left(x\right){|\psi |}^{p-1}\psi =0$, $\left(t,x\right)\in {ℝ}_{+}×{ℝ}^{N}$. Let $G\left(x\right)={\left[V\left(x\right)\right]}^{\frac{p+1}{p-1}-\frac{N}{2}}{\left[K\left(x\right)\right]}^{-\frac{2}{p-1}}$ and suppose that $G\left(x\right)$ has k local minimum points. Then, for any $l\in \left\{1,\cdots ,k\right\}$, we prove the existence of the standing waves in ${H}^{1}\left({ℝ}^{N}\right)$ having exactly l local maximum points which concentrate near l local minimum points of $G\left(x\right)$ respectively as $ϵ\to 0$. The potentials $V\left(x\right)$ and $K\left(x\right)$ are allowed to be either compactly supported or unbounded at infinity. Therefore, we give a positive answer to a problem proposed by Ambrosetti and Malchiodi (2007) [2].

DOI : https://doi.org/10.1016/j.anihpc.2010.05.003
Classification : 35J20,  35J60
Mots clés : Bound state, Multi-peak solutions, Nonlinear Schrödinger equation, Potentials compactly supported or unbounded at infinity
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author = {Ba, Na and Deng, Yinbin and Peng, Shuangjie},
title = {Multi-peak bound states for {Schr\"odinger} equations with compactly supported or unbounded potentials},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {1205--1226},
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Ba, Na; Deng, Yinbin; Peng, Shuangjie. Multi-peak bound states for Schrödinger equations with compactly supported or unbounded potentials. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 5, pp. 1205-1226. doi : 10.1016/j.anihpc.2010.05.003. http://archive.numdam.org/articles/10.1016/j.anihpc.2010.05.003/

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